Methods of modeling physical properties of chemical mixtures and articles of use

ABSTRACT

Included are methods for modeling at least one physical property of a mixture of at least two chemical species. One or more chemical species of the mixture are approximated or represented by at least one conceptual segment. The conceptual segments are then used to compute at least one physical property of the mixture. An analysis of the computed physical properties forms a model of at least one physical property of the mixture. Also included are computer program products and computer systems for implementing the modeling methods.

BACKGROUND OF THE INVENTION

Modeling physical properties of chemical mixtures is an important taskin many industries and processes. Specifically, for many processes,accurate modeling of physical properties for various mixtures is crucialfor such areas as process design and process control applications. Forexample, modeling physical properties of chemical mixtures is oftenuseful when selecting suitable solvents for use in chemical processes.

Solvent selection is an important task in the chemical synthesis andrecipe development phase of the pharmaceutical and agricultural chemicalindustries. The choice of solvent can have a direct impact on reactionrates, extraction efficiency, crystallization yield and productivity,etc. Improved solvent selection brings benefits, such as faster productseparation and purification, reduced solvent emission and lesser waste,lower overall costs, and improved production processes.

In choosing a solvent, various phase behavior characteristics of thesolvent-solute mixtures are considered. For example, vapor-liquidequilibrium (VLE) behavior is important when accounting for the emissionof solvent from reaction mixtures, and liquid-liquid miscibility (LLE)is important when a second solvent is used to extract target moleculesfrom the reaction media. For solubility calculations, solid-liquidequilibrium (SLE) is a key property when product isolation is donethrough crystallization at reduced temperature or with the addition ofanti-solvent.

For many applications, hundreds of typical solvents, not to mention analmost infinite number of mixtures thereof, are candidates in thesolvent selection process. In most cases, there is simply insufficientphase equilibrium data on which to make an informed solvent selection.For example, in pharmaceutical applications, it is often the case thatphase equilibrium data involving new drug molecules in the solventssimply do not exist. Although limited solubility experiments may betaken as part of the trial and error process, solvent selection islargely dictated by researchers' preferences or prior experiences.

Many solubility estimation techniques have been used to model thesolubility of components in chemical mixtures. Some examples include theHansen model and the UNIFAC group contribution model. Unfortunately,these models are rather inadequate because they have been developedmainly for petrochemicals with molecular weights in the 10s and the low100s daltons. These models do not extrapolate well for chemicals withlarger molecular weights, such as those encountered in pharmaceuticalapplications. Pharmaceuticals are mostly large, complex molecules withmolecular weight in the range of about 200-600 daltons.

Perhaps, the most commonly used methods in solvent selection process arethe solubility parameter models, i.e., the regular solution theory andthe Hansen solubility parameter model. There are no binary parameters inthese solubility parameter models and they all follow merely anempirical guide of “like dissolves like.” The regular solution model isapplicable to nonpolar solutions only, but not for solutions where polaror hydrogen-bonding interactions are significant. The Hansen modelextends the solubility parameter concept in terms of three partialsolubility parameters to better account for polar and hydrogen-bondingeffects.

In his book, Hansen published the solubility parameters for over 800solvents. See Hansen, C. M., MANSEN, SOLUBILITY PARAMETERS: A USER'SHANDBOOK (2000). Since Hansen's book contains the parameters for mostcommon solvents, the issue in using the Hansen model lies in thedetermination of the Hansen solubility parameters from regression ofavailable solubility data for the solute of interest in the solventselection process. Once determined, these Hansen parameters provide abasis for calculating activity coefficients and solubilities for thesolute in all the other solvents in the database. For pharmaceuticalprocess design, Bakken, et al. reported that the Hansen model can onlycorrelate solubility data with ±200% in accuracy, and it offers littlepredictive capability. See Bakken, et al., Solubility Modeling inPharmaceutical Process Design, paper presented at AspenTech User GroupMeeting, New Orleans, La., Oct. 5-8, 2003, and Paris, France, Oct.19-22, 2003.

When there are no data available, the UNIFAC functional groupcontribution method is sometimes used for solvent selection. Incomparison to the solubility parameter models, UNIFAC's strength comeswith its molecular thermodynamic foundation. It describes liquid phasenonideality of a mixture with the concept of functional groups. Allmolecules in the mixture are characterized with a set of pre-definedUNIFAC functional groups. The liquid phase nonideality is the result ofthe physical interactions between these functional groups and activitycoefficients of molecules are derived from those of functional groups,i.e., functional group additivity rule. These physical interactions havebeen pre-determined from available phase equilibrium data of systemscontaining these functional groups. UNIFAC gives adequate phaseequilibrium (VLE, LLE and SLE) predictions for mixtures with smallnonelectrolyte molecules as long as these molecules are composed of thepre-defined set of functional groups or similar groups.

UNIFAC fails for systems with large complex molecules for which eitherthe functional group additivity rule becomes invalid or due to undefinedUNIFAC functional groups. UNIFAC is also not applicable to ionicspecies, an important issue for pharmaceutical processes. Anotherdrawback with UNIFAC is that, even when valuable data become available,UNIFAC cannot be used to correlate the data. For pharmaceutical processdesign, Bakken et al., reported that the UNIFAC model only predictssolubilities with a RMS (root mean square) error on ln x of 2, or about±500% in accuracy, and it offers little practical value. Id.

A need exists for new, simple, and practical methods of accuratelymodeling one or more physical properties of a mixture.

SUMMARY OF THE INVENTION

The present invention provides a new system and method for modeling thephysical properties or behavior of chemical mixtures (e.g., chemicalsolutions or suspensions). Briefly, the molecular structure of one ormore species in a chemical mixture is assigned one or more differenttypes of “conceptual segments.” An equivalent number is determined foreach conceptual segment. This conceptual segment approach of the presentinvention is referred to as the Non-Random Two-Liquid Segment ActivityCoefficient (“NRTL-SAC”) model.

In some embodiments, this invention features a method of modeling atleast one physical property of a mixture of at least two chemicalspecies. In one embodiment, the method comprises the computerimplemented steps of determining at least one conceptual segment foreach of the chemical species, using the determined conceptual segmentsto compute at least one physical property of the mixture; and providingan analysis of the computed physical property. The step of determiningat least one conceptual segment includes defining an identity and anequivalent number of each conceptual segment. The provided analysisforms a model of at least one physical property of the mixture.

In further embodiments, this invention includes a method of modeling atleast one physical property of a mixture that includes at least threechemical species. In one embodiment, the method comprises the computerimplemented steps of determining at least one conceptual segment for afirst chemical species; determining at least one conceptual segment fora second chemical species; determining at least one conceptual segmentfor a third chemical species; using the determined conceptual segmentsfor the first chemical species, the determined conceptual segments forthe second chemical species, and the determined conceptual segments forthe third chemical species to compute at least one physical property ofthe mixture; and providing an analysis of the computed physicalproperty. For each conceptual segment, the steps of determining theconceptual segments include defining an identity and an equivalentnumber of the respective conceptual segment. The analysis forms a modelof at least one physical property of the mixture.

In another embodiment, this invention features methods of modelingsolubility of a pharmaceutical component of a mixture that includes atleast one pharmaceutical component and at least one solvent. In oneembodiment, the method comprises the computer implemented steps ofdetermining at least one conceptual segment for the pharmaceuticalcomponent, determining at least one conceptual segment for the solvent,using the determined conceptual segment for the pharmaceutical componentand the determined conceptual segment for the solvent to computesolubility of the pharmaceutical component in the mixture, and providingan analysis of the computed solubility. The steps of determining theconceptual segments include defining an identity and an equivalentnumber of the respective conceptual segment. The analysis forms asolubility model of the pharmaceutical component in the mixture.

In further embodiments, this invention features computer programproducts. In one embodiment, the computer program product comprises acomputer usable medium and a set of computer program instructionsembodied on the computer useable medium for modeling at least onephysical property of a mixture of at least two chemical species. Thecomputer program instructions include the instructions to determine atleast one conceptual segment for each of the chemical species, use thedetermined conceptual segments to compute at least one physical propertyof the chemical mixture; and provide an analysis of the computedphysical property. The program instructions for determining conceptualsegments include instructions for defining an identity and an equivalentnumber of each conceptual segment. The analysis forms a model of atleast one physical property of the mixture.

In yet a further embodiment, this invention features a computer systemfor modeling at least one physical property of a mixture of at least twochemical species. In one embodiment, the computer system comprises auser input means for determining chemical data from a user, a digitalprocessor coupled to receive input (determined chemical data) from theinput means, and an output means coupled to the digital processor. Thedigital processor hosts and executes a modeling system in workingmemory. The modeling system (i) uses the chemical data to determine atleast one conceptual segment for each of the chemical species; (ii) usesthe determined conceptual segments to compute at least one physicalproperty of the chemical mixture, and; (iii) provides an analysis of thecomputed physical property. The modeling system determines a conceptualsegment, in part, by defining an identity and an equivalent number ofeach conceptual segment. The analysis forms a model of at least onephysical property of the mixture. The output means provides to the userthe formed model of the physical property of the chemical mixture.

In some embodiments, this invention features a pharmaceutical compoundmanufactured by a process that includes a modeling method. The modelingmethod models at least one physical property of a mixture of at leasttwo chemical species and comprises the computer implemented steps ofdetermining at least one conceptual segment for each of the chemicalspecies, using the determined conceptual segments to compute at leastone physical property of the mixture; and providing an analysis of thecomputed physical property. The step of determining at least oneconceptual segment includes defining an identity and an equivalentnumber of each conceptual segment. The provided analysis forms a modelof at least one physical property of the mixture.

This invention provides for the fast, practical modeling of physicalproperties or behaviors of chemical mixtures, even when there is littleor no experimental data to which the behavior of the mixture can becorrelated. The formed models offer improved accuracy over most or allprior modeling methods. For example, this invention offers a simple andpractical tool for practitioners to estimate solubility of variouscomponents of a chemical mixture (e.g., a mixture including apharmaceutical component), even when there is little or no phaseequilibrium data available for the mixture.

This invention provides for modeling of mixtures having significanthydrophobic interactions, significant polar interactions, and/orsignificant hydrogen-bonding interactions. This invention eliminates theneed to characterize mixture constituents with sets of pre-definedfunctional groups and provides for the modeling of mixtures comprisinglarge, complex molecules for which a functional group additivity rulebecomes invalid and/or for which there are a number of un-definedfunctional groups.

BRIEF DESCRIPTION OF THE DRAWINGS

The foregoing and other objects, features and advantages of theinvention will be apparent from the following more particulardescription of preferred embodiments of the invention, as illustrated inthe accompanying drawings in which like reference characters refer tothe same parts throughout the different views. The drawings are notnecessarily to scale, emphasis instead being placed upon illustratingthe principles of the invention.

FIG. 1 is a block diagram of a computer system embodying the presentinvention modeling methods.

FIG. 2 illustrates data flow and process steps for a modeler of thepresent invention, such as that employed in the embodiment of FIG. 1.

FIG. 3 illustrates data flow and process steps for a computation by themodeler of FIG. 2.

FIG. 4 illustrates a graph showing the binary phase diagram for a water,1,4-dioxane mixture at atmospheric pressure.

FIG. 5 illustrates a graph showing the binary phase diagram for a water,octanol mixture at atmospheric pressure.

FIG. 6 illustrates a graph showing the binary phase diagram for anoctanol, 1,4-dioxane mixture at atmospheric pressure.

FIG. 7 illustrates a graph showing data of experimental solubilities vs.calculated solubilities for p-aminobenzoic acid in various solvents at298.15K.

FIG. 8 illustrates a graph showing data of experimental solubilities vs.calculated solubilities for benzoic acid in various solvents at 298.15K.

FIG. 9 illustrates a graph showing data of experimental solubilities vs.calculated solubilities for camphor in various solvents at 298.15K.

FIG. 10 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for ephedrine in various solvents at298.15K.

FIG. 11 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for lidocaine in various solvents at298.15K.

FIG. 12 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for methylparaben in various solvents at298.15K.

FIG. 13 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for testosterone in various solvents at298.15K.

FIG. 14 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for theophylline in various solvents at298.15K.

FIG. 15 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for estriol in nine solvents at 298.15K.

FIG. 16 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for estrone in various solvents at 298.15K.

FIG. 17 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for morphine in six solvents at 308.15K.

FIG. 18 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for piroxicam in 14 solvents at 298.15K.

FIG. 19 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for hydrocortisone in 11 solvents at298.15K.

FIG. 20 illustrates a graph showing data of experimental solubilitiesvs. calculated solubilities for haloperidol in 13 solvents at 298.15K.

DETAILED DESCRIPTION OF THE INVENTION

A description of preferred embodiments of the invention follows.

The NRTL-SAC model of the present invention follows the segmentcontribution concept that was first incorporated into the NRTL model asa Gibbs energy expression for oligomers and polymers. While the UNIFACmodel of the prior art decomposes molecules into a large number ofpre-defined functional groups, the NRTL-SAC model of the presentinvention decomposes or assigns to each molecular species a fewpre-defined conceptual segments. For example, in some embodiments of thepresent invention, each molecular species is assigned three types ofconceptual segments: a hydrophobic segment, a polar segment, and ahydrophilic segment. Each conceptual segment is then assigned anequivalent number. The equivalent numbers of these conceptual segmentsare determined, not from their exact molecular structure (as are thefunctional groups of the UNIFAC model), but from experimental data thatreflect on their true molecular characteristics in the mixture. Theseequivalent numbers are used to describe or model how the variousmolecular species of a mixture interact with one another. In thismanner, the NRTL-SAC methods of the present invention is able to modelone or more physical properties of a mixture.

Various NRTL models have been used to model various types of mixtures.Previous segment-based NRTL models used “segments” to define the variouschemical species of a mixture. Like the UNIFAC model, these segmentswere based upon the actual molecular structure of the various chemicalspecies, while the conceptual segments of the present invention aredefined based upon actual thermodynamic behavior of the various chemicalspecies.

The segment contribution approach represents a practical alternative tothe UNIFAC functional group contribution approach. Industrialpractitioners generally have a healthy distrust or suspicion of“predictive” models, empirical or ab initio. Wherever possible, theyprefer correlative models that allow them to validate the model withavailable data, determine the model parameters from the data, andextrapolate into new conditions with proper molecular insights andthermodynamic consistency. The NRTL-SAC model of the present inventionoffers such a framework, with molecular descriptors identified by usingavailable experimental data for the chemical species of a mixture. TheNRTL-SAC model is used to extrapolate to other chemical systems that arealso described in terms of the same or similar set of moleculardescriptors.

In some embodiments, this invention includes methods of modeling atleast one physical property of a mixture of at least two chemicalspecies. In one embodiment, the method comprises the computerimplemented steps of (i) determining at least one conceptual segment foreach of the chemical species; (ii) using the determined conceptualsegments, computing at least one physical property of the mixture; and(iii) providing an analysis of the computed physical property. The stepof determining a conceptual segment includes defining an identity and anequivalent number of each conceptual segment. The analysis forms a modelof at least one physical property of the mixture.

The methods of this invention can model mixtures that include one ormore liquid phases. In some embodiments, at least a portion of at leastone chemical species of the mixture is in at least one fluid phase(e.g., a vapor phase and/or a liquid phase). For example, the mixturecan include one or more liquid phases (e.g., two or more liquid solventphases) and a vapor phase. In further embodiments, at least a portion ofat least one chemical species of the mixture is in one or more solidphases. In yet further embodiments, the mixture includes at least onesolid phase and at least one liquid phase. In still further embodiments,the mixture includes at least one solid phase (e.g., at least 1, 2, 3,or more than 3 solid phases), at least one liquid phase (e.g., at least1, 2, 3, or more than 3 liquid phases), and a vapor phase.

The methods of this invention can model a wide range of chemicalmixtures. For example, the chemical mixtures can include one or more ofthe following types of chemical species: an organic nonelectrolyte, anorganic salt, a compound possessing a net charge, a zwitterions, a polarcompound, a nonpolar compound, a hydrophilic compound, a hydrophobiccompound, a petrochemical, a hydrocarbon, a halogenated hydrocarbon, anether, a ketone, an ester, an amide, an alcohol, a glycol, an amine, anacid, water, an alkane, a surfactant, a polymer, and an oligomer.

In further embodiments, the mixture includes at least one chemicalspecies which is a solvent (e.g., a solvent used in a pharmaceuticalproduction, screening, or testing process), a solute, a pharmaceuticalcomponent, a compound used in an agricultural application (e.g., aherbicide, a pesticide, or a fertilizer) or a precursor of a compoundused in an agricultural application, a compound used in an adhesivecomposition or a precursor of a compound used in an adhesivecomposition, a compound used in an ink composition or a precursor of acompound used in an ink composition. As used herein, a “pharmaceuticalcomponent” includes a pharmaceutical compound, drug, therapeutic agent,or a precursor thereof (i.e., a compound used as an ingredient in apharmaceutical compound production process). In some embodiments, themixture includes at least one pharmaceutical component having amolecular weight greater than about 900 daltons, at least onepharmaceutical component having a molecular weight in the range ofbetween about 100 daltons and about 900 daltons, and/or at least onepharmaceutical component having a molecular weight in the range ofbetween about 200 daltons and about 600 daltons. In further embodiments,the mixture includes at least one nonpolymeric pharmaceutical component.

In further embodiments, the mixture includes at least one ICH solvent,which is a solvent listed in the ICH Harmonized Tripartite Guideline,Impurities: Guideline for Residual Solvents Q3C, incorporated herein inits entirety by reference. ICH STEERING COMMITTEE, ICH HarmonizedTripartite Guideline, Impurities: Guideline for Residual Solvents Q3C,International Conference of Harmonization of Technical Requirements forRegistration of Pharmaceuticals for Human Use (1997).

It will be apparent to those skilled in the art that a component of themixture can belong to more than one type of chemical species.

In accordance with one aspect of the present invention, at least oneconceptual segment (e.g., at least 1, 2, 3, 4, 5, 7, 10, 12, or morethan 12 conceptual segments) is determined or defined for each of thechemical species of the mixture. The conceptual segments are moleculardescriptors of the various molecular species in the mixture. An identityand an equivalent number are determined for each of the conceptualsegments. Examples of identities for conceptual segments include ahydrophobic segment, a polar segment, a hydrophilic segment, a chargedsegment, and the like. Experimental phase equilibrium data can be usedto determine the equivalent number of the conceptual segment(s).

The determined conceptual segments are used to compute at least onephysical property of the mixture, and an analysis of the computedphysical property is provided to form a model of at least one physicalproperty of the mixture. The methods of this invention are able to modela wide variety of physical properties. Examples of physical propertiesinclude vapor pressure, solubility (e.g., the equilibrium concentrationof one or more chemical species in one or more phases of the mixture),boiling point, freezing point, octanol/water partition coefficient,lipophilicity, and other physical properties that are measured ordetermined for use in the chemical processes.

In some embodiments, the mixture includes at least two liquid phases andthe modeled physical property or properties include the solubility ofone or more chemical species in the two liquid phases. In otherembodiments, the mixture includes at least one liquid phase and at leastone solid phase, and the modeled physical property or properties includethe solubility of a chemical species of the solid phase in the liquidphase.

Preferably, the methods provide equilibrium values of the physicalproperties modeled. For example, a mixture can include at least oneliquid solvent and at least one solid pharmaceutical component and themethods can be used to model the solubility of the pharmaceuticalcomponent. In this way, the method can provide the concentration of theamount (e.g., a concentration value) of the pharmaceutical componentthat will be dissolved in the solvent at equilibrium. In anotherexample, the method could model a mixture that includes a solid phase(e.g., a solid pharmaceutical component) and at least two liquid phases(e.g., two solvent that are immiscible in one another). The model canpredict, or be used to predict, how much of the pharmaceutical componentwill be dissolved in the two liquid phases and how much will be left inthe solid phase at equilibrium. In yet a further embodiment, the methodscan be used to predict the behavior of a mixture after a change hasoccurred. For example, if the mixture includes two liquid phases and onesolid phase, and an additional chemical species is introduced into themixture (e.g., a solvent, pharmaceutical component, or other chemicalcompound), additional amounts of a chemical species are introduced intothe mixture, and/or one or more environmental conditions are changes(e.g., a change in temperature and/or pressure), the method can be usedto predict how the introduction of the chemical species and/or change inconditions will alter one or more physical properties of the mixture atequilibrium.

The models of the physical property or properties of the mixture areproduced by determining the interaction characteristics of theconceptual segments. In some embodiments, the segment-segmentinteraction characteristics of the conceptual segments are representedby their corresponding binary NRTL parameters. Given the NRTL parametersfor the conceptual segments and the molecular descriptors for themolecules, the NRTL-SAC model computes activity coefficients for thesegments and then for the various molecules in the mixture. In otherwords, the physical properties or behavior of the mixture will beaccounted for based on the segment compositions of the molecules andtheir mutual interactions. The activity coefficient of each molecule iscomputed from the number and type of segments for each molecule and thecorresponding segment activity coefficients.

In further embodiments, this invention includes a method of modeling atleast one physical property of a mixture that includes at least threechemical species. In one embodiment, the method comprises the computerimplemented steps of (i) determining at least one conceptual segment fora first chemical species; (ii) determining at least one conceptualsegment for a second chemical species; (iii) determining at least oneconceptual segment for a third chemical species; (iv) using thedetermined conceptual segments for the first chemical species, thedetermined conceptual segments for the second chemical species and thedetermined conceptual segments for the third chemical species (e.g., apharmaceutical component), computing at least one physical property ofthe mixture; and (v) providing an analysis of the computed physicalproperty. Each step of determining the conceptual segments includesdefining an identity and an equivalent number of the respectiveconceptual segment. The analysis forms a model of at least one physicalproperty of the mixture.

In further embodiments, this invention features methods of modelingsolubility of a pharmaceutical component of a mixture that includes atleast one pharmaceutical component and at least one solvent. The methodscomprise the computer implemented steps of (i) determining at least oneconceptual segment for the pharmaceutical component; (ii) determining atleast one conceptual segment for the solvent; (iii) using the determinedconceptual segment for the pharmaceutical component and the determinedconceptual segment for the solvent, computing solubility of thepharmaceutical component in the mixture; and (iv) providing an analysisof the computed solubility. The analysis forms a solubility model of thepharmaceutical component in the mixture.

In some embodiments, this invention features computer program products.The computer program products comprise a computer usable medium and aset of computer program instructions embodied on the computer useablemedium for modeling at least one physical property of a mixture of atleast two chemical species. Included are (a) instructions to determineat least one conceptual segment for each of the chemical species; (b)instructions to use the determined conceptual segments to compute atleast one physical property of the chemical mixture; and (c)instructions to provide an analysis of the computed physical property,wherein the analysis forms a model of at least one physical property ofthe mixture.

Referring now to FIG. 1, illustrated is a computer system 10 embodyingthe present invention modeling methods mentioned above. Generally,computer system 10 includes digital processor 12 which hosts andexecutes modeler 20. Modeler 20 comprises the modeling method of theinvention in working memory. Input means 14 provides userselectable/definable chemical data (e.g., data relating to, or usefulfor, modeling a mixture that includes a pharmaceutical component) from auser of computer system 10. Input means 14 can be implemented as any ofvarious in-put/out-put devices, programs, or routines coupled tocomputer system 10.

Responsive to input means 14 is user interface 22. User interface 22receives user input data from input means 14 and provides input data forprocessing by modeler 20. Modeler 20 determines at least one physicalproperty of a mixture that includes at least one user input compound.Modeler 20 further provides an analysis of the determined physicalproperties and thus outputs a model 16 of the determined physicalproperty. As such, output 16 is a model of at least one physicalproperty of a mixture (e.g., a mixture including one or morepharmaceutical components) derived based on the chemical data from input14.

In one embodiment, computer program product 80, including a computerreadable medium (e.g., a removable storage medium such as one or moreDVD-ROM's, CD-ROM's, diskettes, tapes, etc.) provides at least a portionof the software instructions for modeler 20, user interface 22, and/orany of component of modeler 20 or user interface 22. Computer programproduct 80 can be installed by any suitable software installationprocedure, as is well known in the art. In another embodiment, at leasta portion of the software instructions may also be downloaded over awireless connection. Computer program propagated signal product 82embodied on a propagated signal on a propagation medium (e.g., a radiowave, an infrared wave, a laser wave, a sound wave, or an electricalwave propagated over a global network such as the Internet, or othernetwork(s)) provides at least a portion of the software instructions formodeler 20, user interface 22, and/or any component of modeler 20 oruser interface 22.

In alternate embodiments, the propagated signal is an analog carrierwave or digital signal carried on the propagated medium. For example,the propagated signal may be a digitized signal propagated over a globalnetwork (e.g., the Internet), a telecommunications network, or othernetwork. In one embodiment, the propagated signal is a signal that istransmitted over the propagation medium over a period of time, such asthe instructions for a software application sent in packets over anetwork over a period of milliseconds, seconds, minutes, or longer. Inanother embodiment, the computer readable medium of computer programproduct 80 is a propagation medium that the computer system 10 mayreceive and read, such as by receiving the propagation medium andidentifying a propagated signal embodied in the propagation medium, asdescribed above for computer program propagated signal product 82.

FIGS. 2 and 3 illustrate data flow and process steps for a modelerperforming the methods of the invention, such as modeler 20 of FIG. 1.With reference to FIG. 2, chemical data describing one or more chemicalspecies of the mixture and/or environmental conditions (e.g., pressureand/or temperature) is entered at step 105 of the modeler process. Step110 uses that data to determine at least one conceptual segment for eachchemical species of the mixture. The determined conceptual segments areused to compute at least one physical property of the mixture duringstep 115. The computed physical properties are analyzed to form a modelof at least one physical property of the mixture (e.g., solubility ofone or more chemical species in one or more phases of the mixture) instep 120. The model information is then given as output at step 125. Theoutput can take the form of data or an analysis appearing on a computermonitor, data or instructions sent to a process control system ordevice, data entered into a data storage device, and/or data orinstructions relayed to additional computer systems or programs.

FIG. 3 illustrates in more detail the computation at step 115 in FIG. 2.Step 115 begins with the receipt of determined conceptual segments foreach chemical species of the mixture. The determined conceptual segmentsand the equation:

${\ln\;\gamma_{m}^{lc}} = {\frac{\sum\limits_{j}{x_{j}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k}G_{km}}} + {\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime}}G_{{mm}^{\prime}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}( {\tau_{{mm}^{\prime}} - \frac{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}\tau_{{km}^{\prime}}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}} )}}}$are used to compute at least one physical property of the mixture duringstep 215. The computed physical properties are provided as output 220from computation step 215. In step 220, the computed physical propertiesare passed to step 120 of FIG. 2 for forming a model of the physicalproperty of the mixture as described above.

According to the foregoing, in some embodiments, the invention featuresa computer system for modeling at least one physical property of amixture of at least two chemical species. The computer system is formedof a user input means for determining chemical data from a user, adigital processor coupled to receive input from the input means, and anoutput means coupled to the digital processor. The digit processor hostsand executes a modeling system in working memory. The modeling system(i) uses the chemical data to determine at least one conceptual segmentfor each of the chemical species; (ii) uses the determined conceptualsegments to compute at least one physical property of the chemicalmixture; and (iii) provides an analysis of the computed physicalproperty. The analysis forms a model of the at least one physicalproperty of the mixture. The output means provides to the user of theformed model of the physical property of the chemical mixture.

In some embodiments, this invention features a pharmaceutical compoundmanufactured by a process that includes a modeling method. The modelingmethod models at least one physical property of a mixture of at leasttwo chemical species and comprises the computer implemented steps ofdetermining at least one conceptual segment for each of the chemicalspecies, using the determined conceptual segments to compute at leastone physical property of the mixture; and providing an analysis of thecomputed physical property. The step of determining at least oneconceptual segment includes defining an identity and an equivalentnumber of each conceptual segment. The provided analysis forms a modelof at least one physical property of the mixture.

The following Examples are illustrative of the invention, and are notmeant to be limiting in any way.

Example 1 Modeling a Mixture of Nonelectrolyte Chemical Species

A study was performed to determine how well the NRTL-SAC models thesolubility of mixtures comprising a solid organic nonelectrolyte.

The solubility of a solid organic nonelectrolyte is described well bythe expression:

${\ln\; x_{I}^{SAT}} = {{\frac{\Delta_{fus}S}{R}( {1 - \frac{T_{m}}{T}} )} - {\ln\;\gamma_{I}^{SAT}}}$for T≦T_(m) and where the entropy of fusion of the solid (Δ_(fus)S) isrepresented by:

${\Delta_{fus}S} = \frac{\Delta_{fus}H}{T_{m}}$x_(I) ^(SAT) is the mole fraction of the solid (the solute) dissolved inthe solvent phase at saturation, γ_(I) ^(SAT) is the activitycoefficient for the solute in the solution at saturation, R is the gasconstant, T is the temperature, and T_(m) is the melting point of thesolid. Given a polymorph, Δ_(fus)S and T_(m) are fixed and thesolubility is then a function of temperature and activity coefficient ofthe solute in the solution. The activity coefficient of the solute inthe solution plays the key role in determining the solubility. Ingeneral, the activity coefficient of the solute in the solution isusually calculated from a liquid activity coefficient model.

Except for the ideal solution model, an activity coefficient model isoften written in two parts as such:ln γ_(I)=ln γ_(I) ^(C)+ln γ_(I) ^(R)γ_(I) ^(C) and γ_(I) ^(R) are the combinatorial and residualcontributions to the activity coefficient of component I, respectively.

In NRTL-SAC, the combinatorial part, γ_(I) ^(C), is calculated from theFlory-Huggins term for the entropy of mixing. The residual part, γ_(I)^(R), is set equal to the local composition (lc) interactioncontribution, γ_(I) ^(lc):

${\ln\;\gamma_{I}^{R}} = {{\ln\;\gamma_{I}^{lc}} = {\sum\limits_{m}{r_{m,I}\lfloor {{\ln\;\gamma_{m}^{lc}} - {\ln\;\gamma_{m}^{{lc},I}}} \rfloor\mspace{14mu}{with}}}}$${{\ln\;\gamma_{m}^{lc}} = {\frac{\sum\limits_{j}{x_{j}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k}G_{km}}} + \mspace{85mu}{\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime}}G_{{mm}^{\prime}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}\{ {\tau_{{mm}^{\prime}} - \frac{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}\tau_{{km}^{\prime}}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}} )}}}},{{\ln\;\gamma_{m}^{{lc},I}} = {\frac{\sum\limits_{j}{x_{j,I}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k,I}G_{km}}} + {\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime},I}G_{{mm}^{\prime}}}{\sum\limits_{k}{x_{k,I}G_{{km}^{\prime}}}}\{ {\tau_{{mm}^{\prime}} - \frac{\sum\limits_{k}{x_{k,I}G_{{km}^{\prime}}\tau_{{km}^{\prime}}}}{\sum\limits_{k}{x_{k,I}G_{{km}^{\prime}}}}} )}}}},{x_{j} = \frac{\sum\limits_{J}{x_{J}r_{j,J}}}{\sum\limits_{I}{\sum\limits_{i}{x_{I}r_{i,I}}}}},{x_{j,I} = \frac{r_{j,I}}{\sum\limits_{j}r_{j,I}}},$where i, j, k, m, m′ are the segment-based species index, I, J are thecomponent index, x_(j) is the segment-based mole fraction of segmentspecies j, and x_(J) is the mole fraction of component J, r_(m,I) is thenumber of segment species m contained in component I, γ_(m) ^(lc) is theactivity coefficient of segment species m, and γ_(m) ^(lc,I) is theactivity coefficient of segment species m contained only in component I.G and τ are local binary quantities related to each other by the NRTLnon-random factor parameter α:G=exp(−ατ)

The equation:

${\ln\;\gamma_{I}^{R}} = {{\ln\;\gamma_{I}^{lc}} = {\sum\limits_{m}{r_{m,I}\lfloor {{\ln\;\gamma_{m}^{lc}} - {\ln\;\gamma_{m}^{{lc},I}}} \rfloor}}}$is a general form for the local composition interaction contribution toactivity coefficients of components in the NRTL-SAC model of the presentinvention. For mono-segment solvent components (S), this equation can besimplified and reduced to the classical NRTL model as follows:

${\ln\;\gamma_{I = S}^{lc}} = {\sum\limits_{m}{r_{m,S}\lfloor {{\ln\;\gamma_{m}^{lc}} - {\ln\;\gamma_{m}^{{lc},S}}} \rfloor\mspace{14mu}{with}}}$${r_{m,S} = 1},{{\ln\;\gamma_{m}^{{lc},S}} = {0.\mspace{14mu}{Therefore}}},{{\ln\;\gamma_{I = S}^{lc}} = {\frac{\sum\limits_{j}{x_{j}G_{jS}\tau_{jS}}}{\sum\limits_{k}{x_{k}G_{ks}}} + {\sum\limits_{m}{\frac{x_{m}G_{Sm}}{\sum\limits_{k}{x_{k}G_{km}}}( {\tau_{Sm} - \frac{\sum\limits_{k}{x_{k}G_{km}\tau_{km}}}{\sum\limits_{k}{x_{k}G_{km}}}} )}}}},$whereG _(jS)=exp(−α_(jS)τ_(jS)), G _(Sj)=exp(−α_(jS)τ_(Sj)).This is the same equation as the classical NRTL model.

Three conceptual segments were defined for nonelectrolyte molecules: ahydrophobic segment, a polar segment, and a hydrophilic segment. Theseconceptual segments qualitatively capture the phase behavior of realmolecules and their corresponding segments. Real molecules in turn areused as reference molecules for the conceptual segments and availablephase equilibrium data of these reference molecules are used to identifyNRTL binary parameters for the conceptual segments. Preferably, thesereference molecules possess distinct molecular characteristics (i.e.,hydrophobic, hydrophilic, or polar) and have abundant, publiclyavailable, thermodynamic data (e.g., phase equilibrium data).

The study was focused on the 59 ICH solvents used in pharmaceuticalprocess design. Water, triethylamine, and n-octanol were alsoconsidered. Table 1 shows these 62 solvents and the solventcharacteristics.

TABLE 1 Common Solvents in Pharmaceutical Process Design Solvent Solvent(Component 1) τ₁₂ ^(a) τ₂₁ ^(a) τ₁₂ ^(b) τ₂₁ ^(b) τ₁₂ ^(c) τ₂₁ ^(c)characteristics ACETIC-ACID 1.365 0.797 2.445 −1.108 Complex ACETONE0.880 0.935 0.806 1.244 Polar ACETONITRILE 1.834 1.643 0.707 1.787 PolarANISOLE Hydrophobic BENZENE 1.490 −0.614 3.692 5.977 Hydrophobic1-BUTANOL −0.113 2.639 0.269 2.870 −2.157 5.843 Hydrophobic/ Hydrophilic2-BUTANOL −0.165 2.149 −0.168 3.021 −1.539 5.083 Hydrophobic/Hydrophilic N-BUTYL-ACETATE 1.430 2.131 Hydrophobic/PolarMETHYL-TERT-BUTYL- −0.148 0.368 1.534 4.263 Hydrophobic ETHERCARBON-TETRACHLORIDE 1.309 −0.850 5.314 7.369 Hydrophobic CHLOROBENZENE0.884 −0.194 4.013 7.026 Hydrophobic CHLOROFORM 1.121 −0.424 3.587 4.954Hydrophobic CUMENE Hydrophobic CYCLOHEXANE −0.824 1.054 6.012 9.519Hydrophobic 1,2-DICHLOROETHANE 1.576 −0.138 3.207 4.284 2.833 4.783Hydrophobic 1,1-DICHLOROETHYLENE Hydrophobic 1,2-DICHLOROETHYLENEHydrophobic DICHLOROMETHANE 0.589 0.325 1.983 3.828 Polar1,2-DIMETHOXYETHANE 0.450 1.952 Polar N,N-DIMETHYLACETAMIDE −0.564 1.109Polar N,N-DIMETHYLFORMAMIDE 1.245 1.636 −1.167 2.044 PolarDIMETHYL-SULFOXIDE −2.139 0.955 Polar 1,4-DIOXANE 1.246 0.097 1.0031.010 Polar ETHANOL 0.533 2.192 −0.024 1.597 Hydrophobic/ Hydrophilic2-ETHOXYETHANOL −0.319 2.560 −1.593 1.853 Hydrophobic/ HydrophilicETHYL-ACETATE 0.771 0.190 0.508 3.828 Hydrophobic/Polar ETHYLENE-GLYCOL1.380 −1.660 Hydrophilic DIETHYL-ETHER −0.940 1.400 1.612 3.103Hydrophobic ETHYL-FORMATE Polar FORMAMIDE Complex FORMIC-ACID −0.340−1.202 Complex N-HEPTANE −0.414 0.398 Hydrophobic N-HEXANE 6.547 10.9496.547 10.949 Hydrophobic ISOBUTYL-ACETATE Polar ISOPROPYL-ACETATE PolarMETHANOL 1.478 1.155 0.103 0.396 Hydrophobic/ Hydrophilic2-METHOXYETHANOL 1.389 −0.566 Hydrophobic/ Hydrophilic METHYL-ACETATE0.715 2.751 Polar 3-METHYL-1-BUTANOL 0.062 2.374 −0.042 3.029 −0.5985.680 Hydrophobic/Hydrophilic METHYL-BUTYL-KETONE Hydrophobic/PolarMETHYLCYCLOHEXANE 1.412 −1.054 Polar METHYL-ETHYL-KETONE −0.036 1.2730.823 2.128 −0.769 3.883 Hydrophobic/Polar METHYL-ISOBUTYL-KETONE 0.9774.868 Hydrophobic/Polar ISOBUTANOL 0.021 2.027 0.592 2.702 −1.479 5.269Hydrophobic/ Hydrophilic N-METHYL-2-PYRROLIDONE −0.583 3.270 −0.2350.437 Hydrophobic NITROMETHANE 1.968 2.556 Polar N-PENTANE 0.496 −0.523Hydrophobic 1-PENTANOL −0.320 2.567 −0.029 3.583 Hydrophobic/Hydrophilic 1-PROPANOL 0.049 2.558 0.197 2.541 Hydrophobic/ HydrophilicISOPROPYL-ALCOHOL 0.657 1.099 0.079 2.032 Hydrophobic/ HydrophilicN-PROPYL-ACETATE 1.409 2.571 Hydrophobic/Polar PYRIDINE −0.665 1.664−0.990 3.146 Polar SULFOLANE 1.045 0.396 Polar TETRAHYDROFURAN 0.6311.981 1.773 0.563 Polar 1,2,3,4- 1.134 −0.631 HydrophobicTETRAHYDRONAPHTHALENE TOLUENE −0.869 1.292 4.241 7.224 Hydrophobic1,1,1-TRICHLOROETHANE 0.535 −0.197 Hydrophobic TRICHLOROETHYLENE 1.026−0.560 Hydrophobic M-XYLENE Hydrophobic WATER 10.949 6.547 HydrophilicTRIETHYLAMINE −0.908 1.285 1.200 1.763 −0.169 4.997 Hydrophobic/Polar1-OCTANOL −0.888 3.153 0.301 8.939 Hydrophobic/ Hydrophilic Wherein: 1.τ₁₂ ^(a) and τ₂₁ ^(a) are NRTL binary τ parameters for systems of thelisted solvents and hexane. NRTL non-random factor parameter, α, isfixed as a constant of 0.2. In these binary systems, solvent iscomponent 1 and hexane component 2. τ's were determined from availableVLE & LLE data. 2. τ₁₂ ^(b) and τ₂₁ ^(b) are NRTL binary τ parametersfor systems of the listed solvents and water. NRTL non-random factorparameter, α, is fixed as a constant of 0.3. In these binary systems,solvent is component 1 and water component 2. τ's were determined fromavailable VLE data. 3. τ₁₂ ^(c) and τ₂₁ ^(c) are NRTL binary τparameters for systems of the listed solvents and water. NRTL non-randomfactor parameter, α, is fixed as a constant of 0.2. In these binarysystems, solvent is component 1 and water component 2. τ's weredetermined from available LLE data.

Hydrocarbon solvents (aliphatic or aromatic), halogenated hydrocarbons,and ethers are mainly hydrophobic. Ketones, esters and amides are bothhydrophobic and polar. Alcohols, glycols, and amines may have bothsubstantial hydrophilicity and hydrophobicity. Acids are complex, withhydrophilicity, polarity, and hydrophobicity.

Also shown in Table 1 are the available NRTL binary parameters (τ) forvarious solvent-water binary systems and solvent-hexane binary systems.Applicants obtained these binary parameters from fitting selectedliterature phase equilibrium data and deliberately ignoring thetemperature dependency of these parameters. These values illustrate therange of values for these binary parameters. Note that many of thebinary parameters are missing, as the phase equilibrium data is notfound in the literature or simply has never been determined for thatsolvent mixture. Also note the sheer number of binary parameters neededfor the prior art NRTL models for even a moderately sized system ofsolvents. For example, to model 60 solvents with the NRTL model, 60×60NRTL binary parameters would be needed.

Table 1 shows that, for the NRTL binary parameters determined from VLEand LLE data for hydrophobic solvent (1)/water (2) binaries, allhydrophobic solvents exhibit similar repulsive interactions with waterand both τ₁₂ and τ₂₁ are large positive values for the solvent-waterbinaries. When the hydrophobic solvents also carry significanthydrophilic or polar characteristics, τ₁₂ becomes negative while τ₂₁retain a large positive value.

Table 1 also illustrates that similar repulsive, but weaker,interactions between a polar solvent (1) and hexane (2), arepresentative hydrophobic solvent. Both τ₁₂ and τ₂₁ are small, positivevalues for the solvent-hexane binaries. The interactions betweenhydrophobic solvents and hexane are weak and the corresponding NRTLbinary parameters are around or less than unity, characteristic ofnearly ideal solutions.

The interactions between polar solvents (1) and water (2) are moresubtle. While all τ₂₁ are positive, τ₁₂ can be positive or negative.This is probably due to different polar molecules exhibiting differentinteractions, some repulsive and others attractive, with hydrophilicmolecules.

Hexane and water were chosen as the reference molecule for hydrophobicsegment and for hydrophilic segment, respectively. The selection ofreference molecule for polar segment requires attention to the widevariations of interactions between polar molecules and water.Acetonitrile was chosen as the reference molecule for a polar segment,and a mechanism was introduced to tune the way the polar segment ischaracterized. The tuning mechanism, as shown in Table 2, allows tuningof the interaction characteristics between the polar segment and thehydrophilic segment. In other words, instead of using only one polarsegment (“Y”), two polar segments (“Y−” and “Y+”) were used. Thedifference between Y− and Y+ is the way they interact with thehydrophilic segment.

The chosen values for the NRTL binary interactions parameters, α and τ,for the three conceptual segments are summarized in Table 2.

TABLE 2 NRTL Binary Parameters for Conceptual Segments in NRTL-SACSegment (1) X (hydrophobic X (hydrophobic Y− (polar Y+ (polar X(hydrophobic segment) segment) segment) segment) segment) Segment (2) Y−(polar Z (hydrophilic Z (hydrophilic Z (hydrophilic Y+ (polar segment)segment) segment) segment) segment) τ₁₂ 1.643 6.547 −2.000 2.000 1.643τ₂₁ 1.834 10.949 1.787 1.787 1.834 α₁₂ = α₂₁ 0.2 0.2 0.3 0.3 0.2

As a first approximation, the temperature dependency of the binaryparameters was ignored.

The binary parameters for the hydrophobic segment (1)—hydrophilicsegment (2) were determined from available liquid-liquid equilibriumdata of hexane-water binary mixture (see Table 1). α was fixed at 0.2because it is the customary value for a for systems that exhibitliquid-liquid separation. Here both τ₁₂ and τ₂₁ are large positivevalues (6.547, 10.950). They highlight the strong repulsive nature ofthe interactions between the hydrophobic segment and the hydrophilicsegment.

Determining a suitable value for α is known in the art. See J. M.PRAUSNITZ, ET AL., MOLECULAR THERMODYNAMICS OF FLUID-PHASE EQUILIBRIA261 (3d ed. 1999).

The binary parameters for the hydrophobic segment (1)—polar segment (2)were determined from available liquid-liquid equilibrium data ofhexane—acetonitrile binary mixture (see Table 1). Again, α was fixed at0.2. Both τ₁₂ and τ₂₁ were small positive values (1.643, 1.834). Theyhighlight the weak repulsive nature of the interactions betweenhydrophobic segment and polar segment.

The binary parameters for the hydrophilic segment (1)—polar segment (2)were determined from available vapor-liquid equilibrium data ofwater—acetonitrile binary mixture (see Table 1). α was fixed at 0.3 forthe hydrophilic segment—polar segment pair because this binary does notexhibit liquid-liquid separation. τ₁₂ was fixed at a positive value(1.787) and τ₂₁ was allowed to vary between −2 and 2. Two types of polarsegments were allowed, Y− and Y+. For Y− polar segment, the values ofτ₁₂ and τ₂₁ were (1.787, −2). For Y+ polar segment, they were (1.787,2). Note that both Y− polar segment and Y+ polar segment exhibited thesame repulsive interactions with hydrophobic segments as discussed inthe previous paragraph. Also, ideal solution was assumed for Y− polarsegment and Y+ polar segment mixtures (i.e., τ₁₂=τ₂₁=0).

Table 2 captures the general trends for the NRTL binary parameters thatwere observed for a wide variety of hydrophobic, polar, and hydrophilicmolecules.

The application of the NRTL-SAC model requires a databank of moleculardescriptors for common solvents used in the industry. In this example,each solvent was described by using up to four molecular descriptors,i.e., X, Y+, Y−, and Z. So, using four molecular descriptors to model asystem of 60 solvents, a set of up to 4×60 molecular descriptors wouldbe used. However, due to the fact that these molecular descriptorsrepresent certain unique molecular characteristics, often only one ortwo molecular descriptors are needed for most solvents. For example,alkanes are hydrophobic and they are well represented withhydrophobicity, X, alone. Alcohols are hybrids of hydrophobic segmentsand hydrophilic segments and they are well represented with X and Z.Ketones, esters, and ethers are polar molecules with varying degrees ofhydrophobic contents. They are well represented by X and Y's. Hence, theneeded set of molecular descriptors can be much smaller than 4×60.

Determination of solvent molecular descriptors involves regression ofexperimental VLE or LLE data for binary systems of interested solventand the above-mentioned reference molecules (i.e., hexane, acetonitrile,and water) or their substitutes. Solvent molecular descriptors are theadjustable parameters in the regression. If binary data is lacking forthe solvent with the reference molecules, data for other binaries may beused as long as the molecular descriptors for the substitute referencemolecules are already identified. In a way, these reference moleculescan be thought of as molecular probes that are used to elucidate theinteraction characteristics of the solvent molecules. These molecularprobes express the interactions in terms of binary phase equilibriumdata.

Table 3 lists the molecular descriptors identified for the commonsolvents in the ICH list.

TABLE 3 Molecular Descriptors for Common Solvents. Solvent name X Y− Y+Z ACETIC-ACID 0.045 0.164 0.157 0.217 ACETONE 0.131 0.109 0.513ACETONITRILE 0.018 0.131 0.883 ANISOLE 0.722 BENZENE 0.607 0.1901-BUTANOL 0.414 0.007 0.485 2-BUTANOL 0.335 0.082 0.355 N-BUTYL-ACETATE0.317 0.030 0.330 METHYL-TERT-BUTYL-ETHER 1.040 0.219 0.172CARBON-TETRACHLORIDE 0.718 0.141 CHLOROBENZENE 0.710 0.424 CHLOROFORM0.278 0.039 CUMENE 1.208 0.541 CYCLOHEXANE 0.892 1,2-DICHLOROETHANE0.394 0.691 1,1-DICHLOROETHYLENE 0.529 0.208 1,2-DICHLOROETHYLENE 0.1880.832 DICHLOROMETHANE 0.321 1.262 1,2-DIMETHOXYETHANE 0.081 0.194 0.858N,N-DIMETHYLACETAMIDE 0.067 0.030 0.157 N,N-DIMETHYLFORMAMIDE 0.0730.564 0.372 DIMETHYL-SULFOXIDE 0.532 2.890 1,4-DIOXANE 0.154 0.086 0.401ETHANOL 0.256 0.081 0.507 2-ETHOXYETHANOL 0.071 0.318 0.237ETHYL-ACETATE 0.322 0.049 0.421 ETHYLENE-GLYCOL 0.141 0.338DIETHYL-ETHER 0.448 0.041 0.165 ETHYL-FORMATE 0.257 0.280 FORMAMIDE0.089 0.341 0.252 FORMIC-ACID 0.707 2.470 N-HEPTANE 1.340 N-HEXANE 1.000ISOBUTYL-ACETATE 1.660 0.108 ISOPROPYL-ACETATE 0.552 0.154 0.498METHANOL 0.088 0.149 0.027 0.562 2-METHOXYETHANOL 0.052 0.043 0.2510.560 METHYL-ACETATE 0.236 0.337 3-METHYL-1-BUTANOL 0.419 0.538 0.314METHYL-BUTYL-KETONE 0.673 0.224 0.469 METHYLCYCLOHEXANE 1.162 0.251METHYL-ETHYL-KETONE 0.247 0.036 0.480 METHYL-ISOBUTYL-KETONE 0.673 0.2240.469 ISOBUTANOL 0.566 0.067 0.485 N-METHYL-2-PYRROLIDONE 0.197 0.3220.305 NITROMETHANE 0.025 1.216 N-PENTANE 0.898 1-PENTANOL 0.474 0.2230.426 0.248 1-PROPANOL 0.375 0.030 0.511 ISOPROPYL-ALCOHOL 0.351 0.0700.003 0.353 N-PROPYL-ACETATE 0.514 0.134 0.587 PYRIDINE 0.205 0.1350.174 SULFOLANE 0.210 0.457 TETRAHYDROFURAN 0.235 0.040 0.3201,2,3,4-TETRAHYDRONAPHTHALENE 0.443 0.555 TOLUENE 0.604 0.3041,1,1-TRICHLOROETHANE 0.548 0.287 TRICHLOROETHYLENE 0.426 0.285 M-XYLENE0.758 0.021 0.316 WATER 1.000 TRIETHYLAMINE 0.557 0.105 1-OCTANOL 0.7660.032 0.624 0.335

Among the ICH solvents, the molecular descriptors identified foranisole, cumene, 1,2-dichloroethylene, 1,2-dimethoxyethane,N,N-dimethylacetamide, dimethyl sulfoxide, ethyl formate, isobutylacetate, isopropyl acetate, methyl-butyl-ketone, tetralin, andtrichloroethylene were questionable, due to lack of sufficientexperimental binary phase equilibrium data. In fact, no public data formethyl-butyl-ketone (2-hexanone) was found and its molecular descriptorswere set to be the same as those for methyl-isobutyl-ketone.

The NRTL-SAC model with the molecular descriptors qualitatively capturesthe interaction characteristics of the solvent mixtures and theresulting phase equilibrium behavior. FIGS. 4 to 6 contain three graphsillustrating the binary phase diagrams for a water, 1,4-dioxane, andoctanol system at atmospheric pressure. The graphs illustrate thepredictions of both the NRTL model with the binary parameters in Table 1and NRTL-SAC models with the model descriptors of Table 3. FIG. 4illustrates the water, 1,4-dioxane mixture; FIG. 5 illustrates thewater, octanol mixture; and FIG. 6 illustrates the octanol, 1,4-dioxanemixture. The predictions with the NRTL-SAC model are broadly consistentwith the calculations from the NRTL model that are generally understoodto represent experimental data within engineering accuracy.

Example 2 Model Prediction Results

Data compiled by Marrero and Abildsko provides a good source ofsolubility data for large, complex chemicals. Marrero, J. & Abildskov,J., Solubility and Related Properties of Large Complex Chemicals, Part1: Organic Solutes Ranging from C ₄ to C ₄₀, CHEMISTRY DATA SERIES XV,DECHEMA, (2003). From that applicants extracted solubility data for the8 molecules reported by Lin and Nash. Lin, H.-M. & R. A. Nash, AnExperimental Method for Determining the Hildebrand Solubility Parameterof Organic Electrolytes, 82 J. PHARMACEUTICAL SCI. 1018 (1993). Alsotested, were 6 additional molecules with sizable solubility data sets.

The NRTL-SAC model was applied to the solvents that are included inTable 3. The molecular descriptors determined for the solutes aresummarized in Table 4. During the data regression, all experimentalsolubility data, regardless of the order of magnitude, were assignedwith a standard deviation of 20%. The comparisons between theexperimental solubility and the calculated solubility are given in FIGS.7 to 20, which illustrate phase diagrams for the systems at 298.15K andatmospheric pressure.

Good representations for the solubility data was obtained with theNRTL-SAC model. The RMS errors in ln x for the fits are given in Table4.

TABLE 4 Molecular descriptors for solutes. RMS # of error on Solute MWsolvents T (K) X Y− Y+ Z lnK_(sp) ln x p-Aminobenzoic 137.14  7 298.150.218 0.681 1.935 0.760 −2.861 0.284 acid Benzoic acid 122.12  7 298.150.524 0.089 0.450 0.405 −1.540 0.160 Camphor 152.23  7 298.15 0.6040.124 0.478 0.000 −0.593 0.092 Ephedrine 165.23  7 298.15 0.458 0.0680.000 0.193 −0.296 0.067 Lidocaine 234.33  7 298.15 0.698 0.596 0.2930.172 −0.978 0.027 Methylparaben 152.14  7 298.15 0.479 0.484 1.2180.683 −2.103 0.120 Testosterone 288.41  7 298.15 1.051 0.771 0.233 0.669−3.797 0.334 Theophylline 180.18  7 298.15 0.000 0.757 1.208 0.341−6.110 0.661 Estriol 288.38  9^(a) 298.15 0.853 0.000 0.291 1.928 −7.6520.608 Estrone 270.37 12 298.15 0.499 0.679 1.521 0.196 −6.531 0.519Morphine 285.34  6 308.15 0.773 0.000 0.000 1.811 −4.658 1.007 Piroxicam331.35 14^(b) 298.15 0.665 0.000 1.803 0.169 −7.656 0.665 Hydrocortisone362.46 11^(c) 298.15 0.401 0.970 1.248 0.611 −6.697 0.334 Haloperidol375.86 13^(d) 298.15 0.827 0.000 0.000 0.131 −4.398 0.311 ^(a)With THFexcluded. ^(b)With 1,2 dichloroethane, chloroform, diethyl ether, andDMF excluded. ^(c)With hexane excluded. ^(d)With chloroform and DMFexcluded. K_(sp), the solubility product constant, corresponds to theideal solubility (in mole fraction) for the solute. The quality of thefit reflects both the effectiveness of the NRTL-SAC model and thequality of the molecular descriptors identified from the limitedavailable experimental data for the solvents.

FIGS. 7, 8, 9, 10, 11, 12, 13, and 14 include graphs illustrating theexperimental solubilities vs. calculated solubilities for p-aminobenzoicacid, benzoic acid, camphor, ephedrine, lidocaine, methylparaben,testosterone, and theophylline, respectively, in various solvents at298.15K. The various solvents used were selected from a group of 33solvents, including acetic acid, acetone, benzene, 1-butanol, n-butylacetate, carbon tetrachloride, chlorobenzene, chloroform, cyclohexane,1,2-dichloroethane, dichloromethane, 1,2-dimethoxyethane,N,N-dimethylformamide, dimethyl-sulfoxide, 1,4-dioxane, ethanol,2-ethoxyethanol, ethyl acetate, ethylene glycol, diethyl ether,formamide, n-heptane, n-hexane, isopropyl acetate, methanol, methylacetate, 1-pentanol, 1-propanol, isopropyl alcohol, teterhydrofuran,toluene, water, and 1-octanol. The experimental solubility data wasrepresented well with the NRTL-SAC model.

FIG. 15 includes a graph illustrating the experimental solubilities vs.calculated solubilities for estriol in 9 solvents at 298.15K. Theexperimental solubility data was represented well with the NRTL-SACmodel. The data for tetrahydrofuran is found to be a very significantoutlier and it is not included in the 9 solvents shown in FIG. 15.

FIG. 16 includes a graph illustrating the experimental solubilities vs.calculated solubilities for estrone in various solvents at 298.15K. Theexperimental solubility data was represented well with the NRTL-SACmodel.

FIG. 17 includes a graph illustrating the experimental solubilities vs.calculated solubilities for morphine in 6 solvents at 308.15K.Cyclohexane and hexane were outliers. They are very low solubilitysolvents for morphine and the quality of the data is possibly subject tolarger uncertainties.

FIG. 18 illustrates a graph of the experimental solubilities vs.calculated solubilities for piroxicam in 14 solvents at 298.15K.1,2-dichloroethane, chloroform, diethyl ether, and N,N-dimethylformamide(DMF) were found to be major outliers and are not included in the 14solvents shown in FIG. 18. Interestingly, Bustamante, et al. alsoreported 1,2-dichloroethane, chloroform, and diethyl ether as outliersin their study based on solubility parameter models. P. Bustamante, etal., Partial Solubility Parameters of Piroxicam and Niflumic Acid, 1998INT. J. OF PHARM. 174, 141.

FIG. 19 illustrates a graph of the experimental solubilities vs.calculated solubilities for hydrocortisone in 11 solvents at 298.15K.Hexane is excluded because of the extreme low solubility ofhydrocortisone in hexane which could possibly subject the data to largeruncertainty.

FIG. 20 illustrates a graph of the experimental solubilities vs.calculated solubilities for haloperidol in 13 solvents at 298.15K.Haloperidol showed unusually high solubilities in chloroform and DMF andthese two solvents are not included in the 13 solvents.

The average RMS error on ln x for the predictions vs. experimentalsolubility data in Table 4 is 0.37. This corresponds to about ±45%accuracy in solubility predictions.

Experiment 3 Comparison of NRTL-SAC Model to Prior Art Methods forPharmaceutical Components

The solubilities of various pharmaceutical compounds was modeled withthe NRTL-SAC approach of the present invention as well as some prior artmodels (e.g., the Hanson model and the UNIFAC model) to compare theirrelative accuracies. The pharmaceutical compounds used included VIOXX®,ARCOXIA®, Lovastatin, Simvastatin, FOSAMAX®. (Available from Merck &Co., Inc., Whitehouse Station, N.J.). The solvents used included water,N,N-Dimethylformamide (“DMF”), 1-propanol, 2-propanol, 1-butanol,toluene, Chloro-benzene, acetonitrile, ethyl acetate, methanol, ethanol,heptane, acetone, and triethylamine (TEA).

Saturated solutions of the compounds in the solvents were allowed toequilibrate for at least 48 hours. Supernatant fluid was filtered anddiluted, and an a high pressure liquid chromatography (HPLC)concentration analysis was performed to compare the predicted solubilityvalues with actual solubility values.

The NRTL-SAC model of the present invention gave a RMS error on ln x ofabout 0.5 (i.e., an accuracy and predictive capability of ±˜50%), whilethe Hansen model had a RMS error on ln x of more than 0.75 and theUNIFAC model had a RMS error on ln x of more than 1.75. Additionalcomparisons were made for dual-solvent/pharmaceutical systems, andacceptable predictions were obtained from the NRTL-SAC model of thepresent invention.

These experiments show that the NRTL-SAC model is a simple correlativeactivity coefficient equation that requires only component-specificmolecular descriptors (i.e., conceptual segments). Conceptually, theapproach suggests that a practitioner account for the liquid ideality ofboth small solvent molecules and complex pharmaceutical molecules interms of component-specific molecular descriptors (e.g., hydrophobicity,polarity, and hydrophilicity). In practice, these molecular descriptorsbecome the adjustable parameters that are determined from selectedexperimental data. With the development of molecular descriptors forsolvents and organic solutes, engineering calculations can be performedfor various phase equilibrium studies, including solubilities insolvents and solvent mixtures for solvent selection. The NRTL-SAC modelprovides good qualitative representation on phase behaviors of organicsolvents and their complex pharmaceutical solutes and it offers apractical predictive methodology for use in pharmaceutical processdesign.

While this invention has been particularly shown and described withreferences to preferred embodiments thereof, it will be understood bythose skilled in the art that various changes in form and details may bemade therein without departing from the scope of the inventionencompassed by the appended claims.

1. A method of modeling at least one physical property of a mixture ofat least two chemical species using a modeler, the method comprising thecomputer implemented steps of: a) providing a computer programmed toserve as a modeler, the modeler during execution being formed of (i) adatabank of molecular descriptors of known chemical species, and (ii) acalculator of molecular descriptors of unknown chemical species, themodeler being configured to be executable by a processor; b) determiningat least one conceptual segment, instead of a molecular structuralsegment, for each of the at least two chemical species, the conceptualsegment being determined from in-mixture behavior of the at least twochemical species, including for each conceptual segment, (i) identifyingthe conceptual segment as one of a hydrophobic segment, a hydrophilicsegment, a polar segment, or a combination thereof, and (ii) defining anequivalent number for the conceptual segment, the equivalent numberbeing based on experimental phase equilibrium data and being one ofcarried in the databank of molecular descriptors of known chemicalspecies or obtained using the calculator of molecular descriptors ofunknown chemical species by regression of experimental phase equilibriumdata for binary systems of unknown chemical species and referencechemical species; c) providing the determined at least one conceptualsegment to the modeler, and in response the modeler using the determinedat least one conceptual segment to compute at least one physicalproperty of the mixture, including any one of vapor pressure,solubility, boiling point, freezing point, octanol/water partitioncoefficient, or a combination thereof, the modeler computing the atleast one physical property by determining an activity coefficient ofone of the at least two chemical species, the activity coefficient beingformed of at least a residual contribution to the activity coefficientof the one chemical species, the modeler setting the residualcontribution equal to a local composition interaction contribution tothe activity coefficient for the one chemical species based on thedetermined at least one conceptual segment; d) analyzing the computed atleast one physical property using the modeler, in a comparison to thecomputed at least one physical property of other mixtures of at leasttwo chemical species, and forming therefrom a model of the at least onephysical property of the mixture; and e) outputting the formed modelfrom the modeler to a computer display monitor.
 2. The method of claim1, wherein the mixture includes more than one phase and at least aportion of at least one of the at least two chemical species is in afluid phase.
 3. The method of claim 2, wherein the mixture includes anynumber and combination of vapor, solid, and liquid phases.
 4. The methodof claim 1, wherein the mixture includes at least one liquid phase. 5.The method of claim 1, wherein the mixture includes at least one liquidphase and at least one solid phase.
 6. The method of claim 1, whereinthe mixture includes at least one liquid solvent and at least onepharmaceutical component.
 7. The method of claim 1, wherein the modeleruses the determined at least one conceptual segment to computelipophilicity.
 8. The method of claim 1, wherein the mixture includesmore than one phase and at least a portion of at least one of the atleast two chemical species forms a vapor phase.
 9. The method of claim8, wherein the mixture includes at least one liquid phase and whereinthe computed at least one physical property includes any of pressure ofthe vapor phase, the solubility of the at least one liquid phase, thesolubility of the vapor phase in at least one liquid phase, or acombination thereof.
 10. The method of claim 1, wherein the mixtureincludes a first liquid phase, a second liquid phase, and a firstchemical species, and wherein at least a portion of the first chemicalspecies is dissolved in both the first liquid phase and the secondliquid phase.
 11. The method of claim 10, wherein analyzing by themodeler provides a solubility model for the mixture.
 12. The method ofclaim 10, wherein the computed at least one physical property includessolubility of at least one of the at least two chemical species in atleast one phase of the mixture.
 13. The method of claim 1, wherein atleast one of the at least two chemical species of the mixture is apolymer.
 14. The method of claim 1, wherein at least one of the at leasttwo chemical species of the mixture is used as a component of anadhesive mixture.
 15. The method of claim 1, wherein at least one of theat least two chemical species of the mixture is used as a component ofan ink mixture.
 16. The method of claim 1, wherein at least one of theat least two chemical species of the mixture is used as a component of amixture having an agricultural application.
 17. The method of claim 1,wherein determining at least one conceptual segment for each of the atleast two chemical species includes determining at least two conceptualsegments for at least one of the at least two chemical species.
 18. Themethod of claim 1, wherein the mixture includes a solid phase, a liquidphase, and a first chemical species, wherein at least a portion of thefirst chemical species is in the solid phase, and the modeler computesat least one physical property of the first chemical species bycalculating:${{\ln\; x_{I}^{SAT}} = {{\frac{\Delta_{fus}S}{R}( {1 - \frac{T_{m}}{T}} )} - {\ln\;\gamma_{I}^{SAT}}}},$wherein T is a temperature of the mixture, T_(m) is the meltingtemperature of the first chemical species in the solid phase of themixture, T is less than or equal to T_(m), x_(I) ^(SAT) is the molefraction of the first chemical species dissolved in the liquid phase atsaturation, Δ_(fus) S is the entropy of fusion of the first chemicalspecies, γ_(I) ^(SAT) is an activity coefficient, γ_(I), for the firstchemical species in the liquid phase at saturation, and R is the gasconstant.
 19. The method of claim 18, wherein the modeler computes theat least one physical property by determining the activity coefficientγ_(I), for the first chemical species such thatln γ_(I)=ln γ_(I) ^(C)+ln γ_(I) ^(R), where γ_(I) ^(C) is acombinatorial contribution to the activity coefficient for the firstchemical species of the mixture, and γ_(I) ^(R) is a residualcontribution to the activity coefficient of the first chemical speciesthat is set equal to a local composition interaction contribution to theactivity coefficient for the first chemical species based on thedetermined at least one conceptual segment.
 20. The method of claim 19,wherein the modeler computes the at least one physical property bycomputing the local composition interaction contribution γ_(I) ^(lc) tothe activity coefficient for the first chemical species as${{\ln\;\gamma_{I}^{R}} = {{\ln\;\gamma_{I}^{lc}} = {\sum\limits_{m}{r_{m,l}\lbrack {{\ln\;\gamma_{m}^{lc}} - {\ln\;\gamma_{m}^{{lc},I}}} \rbrack}}}},{wherein}$${{\ln\;\gamma_{m}^{lc}} = {\frac{\sum\limits_{j}{x_{j}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k}G_{km}}} + \mspace{85mu}{\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime}}G_{{mm}^{\prime}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}( {\tau_{{mm}^{\prime}} - \frac{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}\tau_{{km}^{\prime}}}}{\sum\limits_{k}{x_{k}G_{{km}^{\prime}}}}} )}}}};$${{\ln\;\gamma_{m}^{{lc},I}} = {\frac{\sum\limits_{j}{x_{j,I}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k,I}G_{km}}} + \mspace{101mu}{\sum\limits_{m^{\prime}}{\frac{x_{m^{\prime},I}G_{{mm}^{\prime}}}{\sum\limits_{k}{x_{k,I}G_{{km}^{\prime}}}}( {\tau_{{mm}^{\prime}} - \frac{\sum\limits_{k}{x_{k,I}G_{{km}^{\prime}}\tau_{{km}^{\prime}}}}{\sum\limits_{k}{x_{k,l}G_{{km}^{\prime}}}}} )}}}};$${x_{j} = \frac{\sum\limits_{J}{x_{J}r_{j,J}}}{\sum\limits_{I}{\sum\limits_{i}{x_{I}r_{i,I}}}}};{x_{j,I} = \frac{r_{j,I}}{\sum\limits_{j}r_{j,I}}};$wherein: i, j, k, m, and m′ are conceptual segment species, one of ahydrophobic segment, a hydrophilic segment, a polar segment, or acombination thereof; I and J are chemical species; x_(j) is a conceptualsegment mole fraction of conceptual segment species j; x_(J) is a molefraction of J; r_(m,I) is the equivalent number of the conceptualsegment species m contained in I; γ_(m) ^(lc) is an activity coefficientof conceptual segment species m; γ_(m) ^(lc,I) is an activitycoefficient of conceptual segment species m contained only in I; G and τare local binary quantities related to each other by a non-random factorparameter α; and G=exp(−ατ).
 21. A method of modeling at least onephysical property of a mixture that includes at least three chemicalspecies using a modeler, the method comprising the computer implementedsteps of: a) providing a computer programmed to serve as a modeler, themodeler during execution being formed of (i) a databanik of moleculardescriptors of known chemical species, and (ii) a calculator ofmolecular descriptors of unknown chemical species, the modeler beingconfigured to be executable by a processor; b) determining at least oneconceptual segment, instead of a molecular structural segment, for afirst chemical species, the conceptual segment being determined fromin-mixture behavior of the at least three chemical species, includingfor the at least one conceptual segment, (i) identifying the conceptualsegment as one of a hydrophobic segment, a hydrophilic segment, a polarsegment, or a combination thereof, and (ii) defining for the firstchemical species an equivalent number for the at least one conceptualsegment for the first chemical species, the equivalent number beingbased on experimental phase equilibrium data and being carried in thedatabanik of molecular descriptors of known chemical species or obtainedusing the calculator of molecular descriptors of unknown chemicalspecies by regression of experimental phase equilibrium data for binarysystems of unknown chemical species and reference chemical species; andc) determining at least one conceptual segment, instead of a molecularstructural segment, for a second chemical species, including for the atleast one conceptual segment of the second chemical species, (i)identifying the conceptual segment as one of a hydrophobic segment, ahydrophilic segment, a polar segment, or a combination thereof, and (ii)defining for the second chemical species an equivalent number for the atleast one conceptual segment for the second chemical species, theequivalent number being based on experimental phase equilibrium data andbeing carried in the databanik of molecular descriptors of knownchemical species or obtained using the calculator of moleculardescriptors of unknown chemical species by regression of experimentalphase equilibrium data for binary systems of unknown chemical speciesand reference chemical species; d) determining at least one conceptualsegment, instead of a molecular structural segment, for a third chemicalspecies, including for the at least one conceptual segment of the thirdchemical species, (i) identifying the conceptual segment as one of ahydrophobic segment, a hydrophilic segment, a polar segment, or acombination thereof, and (ii) defining for the third chemical species anequivalent number for the at least one conceptual segment for the thirdchemical species, the equivalent number being based on experimentalphase equilibrium data and being carried in the databanik of moleculardescriptors of known chemical species or obtained using the calculatorof molecular descriptors of unknown chemical species by regression ofexperimental phase equilibrium data for binary systems of unknownchemical species and reference chemical species; e) providing to themodeler the determined at least one conceptual segment for the firstchemical species, the determined at least one conceptual segment for thesecond chemical species and the determined at least one conceptualsegment for the third chemical species, and in response the modelerusing the determined at least one conceptual segment for each of thefirst, second, and third chemical species to compute at least onephysical property of the mixture, including any one of vapor pressure,solubility, boiling point, freezing point, octanol/water partitioncoefficient, or a combination thereof, the modeler computing the atleast one physical property by determining an activity coefficient ofone of the at least three chemical species, the activity coefficientbeing formed of at least a residual contribution to the activitycoefficient of the one chemical species, the modeler setting theresidual contribution equal to a local composition interactioncontribution to the activity coefficient for the one chemical speciesbased on the determined at least one conceptual segment of the onechemical species; and f) analyzing the computed at least one physicalproperty using the modeler, in a comparison to the computed at least onephysical property of other mixtures of at least three chemical species,and forming therefrom a model of the at least one physical property ofthe mixture; and g) outputting the formed model from the modeler to acomputer display monitor.
 22. The method of claim 21, wherein at least aportion of the first chemical species is in a fluid phase, at least aportion of the second chemical species is in the fluid phase, and atleast a portion of the third chemical species is in the fluid phase. 23.The method of claim 21, wherein at least a portion of the first chemicalspecies is in a fluid phase, at least a portion of the second chemicalspecies is in a the fluid phase, and at least a portion of the thirdchemical species is in a solid phase.
 24. The method of claim 21,wherein the modeler uses the determined at least one conceptual segmentto compute lipophilicity.
 25. The method of claim 21, wherein themixture includes a first liquid phase, a second liquid phase, andwherein at least a portion of the third chemical species is dissolved inboth the first liquid phase and the second liquid phase.
 26. The methodof claim 25, wherein the analyzing by the modeler provides a solubilitymodel for the mixture.
 27. The method of claim 25, wherein the computedat least one physical property includes solubility of the first chemicalspecies, the second chemical species, the third chemical species, or acombination thereof in at least one phase of the mixture.
 28. The methodof claim 21, wherein at least two conceptual segments are determined forthe first chemical species, the second chemical species, the thirdchemical species, or a combination thereof.
 29. The method of claim 21,wherein the mixture includes a solid phase and a liquid phase, whereinat least a portion of the one chemical species is in the solid phase,and the modeler computes at least one physical property by calculating:${{\ln\; x_{I}^{SAT}} = {{\frac{\Delta_{fus}S}{R}( {1 - \frac{T_{m}}{T}} )} - {\ln\;\gamma_{I}^{SAT}}}},$wherein: T is a temperature of the mixture; T_(m) is the meltingtemperature of the one chemical species in the solid phase of themixture; T is less than or equal to T_(m); x_(I) ^(SAT) is the molefraction of the one chemical species dissolved in the liquid phase atsaturation; Δ_(fus) S is the entropy of fusion of the one chemicalspecies; γ_(I) ^(SAT) is the activity coefficient, γ_(I), for the onechemical species in the liquid phase at saturation; and R is the gasconstant.
 30. The method of claim 29, wherein the modeler computes theat least one physical property by determining the activity coefficientγ_(I) for the one chemical species, such thatln γ_(I)=ln γ_(I) ^(C)+ln γ_(I) ^(R), where γ_(I) ^(C) is acombinatorial contribution to the activity coefficient for the onechemical species of the mixture, and γ_(I) ^(R) is a residualcontribution to the activity coefficient of the one chemical speciesthat is set equal to the local composition interaction contribution tothe activity coefficient for the one chemical species based on thedetermined at least one conceptual segment of the one chemical species.31. The method of claim 30, wherein the modeler computes the at leastone physical property by computing the local interaction contributionγ_(I) ^(lc) as${{\ln\;\gamma_{l}^{R}} = {{\ln\;\gamma_{l}^{l\; c}} = {\sum\limits_{m}{r_{m,l}\lbrack {{\ln\;\gamma_{m}^{lc}} - {\ln\;\gamma_{m}^{{lc},l}}} \rbrack}}}},{{{{wherein}\mspace{14mu}\ln\;\gamma_{m}^{lc}} = {\frac{\sum\limits_{j}{x_{j}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k}G_{k\; m}}} + {\sum\limits_{m^{\prime}}{\frac{x_{m}^{\prime}G_{m\; m}^{\prime}}{\sum\limits_{k}{x_{k}G_{k\; m}^{\prime}}}( {\tau_{m\; m}^{\prime} - \frac{\sum\limits_{k}{x_{k}G_{k\; m}\tau_{k\; m}^{\prime}}}{\sum\limits_{k}{x_{k}G_{k\; m}^{\prime}}}} )}}}};{{\ln\;\gamma_{m}^{{lc},l}} = {\frac{\sum\limits_{j}{x_{j,l}G_{jm}\tau_{jm}}}{\sum\limits_{k}{x_{k,l}G_{k\; m}}} + {\sum\limits_{m^{\prime}}{\frac{x_{m,l}^{\prime}G_{m\; m}^{\prime}}{\sum\limits_{k}{x_{k,l}G_{k\; m}^{\prime}}}( {\tau_{m\; m^{\prime}} - \frac{\sum\limits_{k}{x_{k,l}G_{k\; m}^{\prime}\tau_{k\; m}^{\prime}}}{\sum\limits_{k}{x_{k,l}G_{k\; m}^{\prime}}}} )}}}};}$${x_{j} = \frac{\sum\limits_{J}{x_{j}r_{J,J}}}{\sum\limits_{I}{\sum\limits_{i}{x_{I}r_{i,l}}}}};{x_{j,l} = \frac{r_{j,I}}{\sum\limits_{j}r_{j,I}}};$wherein: i, j, k, m, and m′ are conceptual segment species, one of ahydrophobic segment, a hydrophilic segment, a polar segment, or acombination thereof I and J are chemical species; x_(j) is a conceptualsegment mole fraction of conceptual segment species j; x_(J) is a molefraction of J; r_(m,I) is the equivalent number of conceptual segmentspecies m contained in I; γ_(m) ^(lc) is an activity coefficient ofconceptual segment species m; γ_(m) ^(lc,I) is an activity coefficientof conceptual segment species m contained only in I; G and τ are localbinary quantities related to each other by a non-random factor parameterα; and G=exp(−ατ).
 32. A computer program product, comprising a computerreadable storage medium having stored thereon a set of computer programinstructions for modeling at least one physical property of a mixture ofat least two chemical species, wherein the computer program instructionswhen executed by a computer cause the computer to: a) provide a modelerformed of a databanik of molecular descriptors of known chemicalspecies, and a calculator of molecular descriptors of unknown chemicalspecies; b) determine at least one conceptual segment, instead of amolecular structural segment, for each of the at least two chemicalspecies, the conceptual segment being determined from in-mixturebehavior of the at least two chemical species, including for eachconceptual segment, (i) identifying the conceptual segment as one of ahydrophobic segment, a hydrophilic segment, a polar segment, or acombination thereof, and (ii) defining an equivalent number for theconceptual segment, the equivalent number being based on experimentalphase equilibrium data and being one of carried in the databanik ofmolecular descriptors of known chemical species or obtained using thecalculator of molecular descriptors of unknown chemical species byregression of experimental phase equilibrium data for binary systems ofunknown chemical species and reference chemical species; c) provide thedetermined at least one conceptual segment in the modeler, to compute atleast one physical property of the chemical mixture including any one ofvapor pressure, solubility, boiling point, freezing point, octanol/waterpartition coefficient, or a combination thereof, wherein the modelercomputes the at least one physical property by determining an activitycoefficient of one of the at least two chemical species, the activitycoefficient being formed of at least a residual contribution to theactivity coefficient of the one chemical species, and wherein themodeler sets the residual contribution equal to a local compositioninteraction contribution to the activity coefficient for the onechemical species based on the determined at least one conceptualsegment; d) analyze the computed at least one physical property usingthe modeler, in a comparison to the computed at least one physicalproperty of other mixtures of at least two chemical species, and formtherefrom a model of the at least one physical property of the mixture;and e) output the formed model from the modeler to a user.
 33. Thecomputer program product of claim 32, wherein at least some portion ofthe computer program instructions include instructions to request dataor request instructions over a telecommunications network.
 34. Thecomputer program product of claim 32, wherein at least some portion ofthe set of computer program instructions is transmitted over a globalnetwork.
 35. The computer program product of claim 32, wherein thecomputer readable medium includes a removable storage medium.
 36. Thecomputer program product of claim 35, wherein the removable storagemedium includes any of a CD-ROM, a DVD-ROM, a diskette, and a tape. 37.The method of claim 1, wherein the in-mixture behavior is determinedfrom any combination of experimental phase equilibrium data orcomputational estimates.